On the nonlinear regularized Schrödinger equation in exterior domains
Let \(\Theta\subset\mathbb R^d\), \(d\in\{2,3\}\), be a compact domain with a smooth boundary. Consider the exterior domain \(\Omega=\mathbb R^d\setminus\Theta\) and the following nonlinear regularized Schrödinger equation \[ (*)\quad\left\{\begin{aligned}i\partial_t u-\Delta_D+i a(x)(-\Delta_D)^{1/2} a(x) u&=F(u)&\quad (t>0,x\in\Omega)\\ u(0,x)&=u_0(x)&\\ u(t,x)&=0&\quad(x\in\Omega)\end{aligned}\right. \] where \(u\) is the unkown function, \(u_0\in C_0^\infty(\Omega), a\in C^\infty(\bar{\Omega})\) are given functions, and \(-\Delta_D\) is the Dirichelt Laplace operator on \(\Omega\). The function \(F\) is non-linear and satisfies \(F(0)=0\). \\ Using the Strichartz estimates from [\textit{M. Karmous}, J. Abstr. Differ. Equ. Appl. 6, No. 1, 37--50 (2015; Zbl 1326.35303)], the authors of the paper under review establish the following two results \begin{itemize} \item[i.] Local well-posedness in \(H^1_0(\Omega)\) for solutions of \((*)\) \item[ii.] Global existence and uniqueness in \(L^2(\Omega)\) for solutions of \((*)\) \end{itemize} For each of the foregoing results i. and ii. to hold, the function \(a\) and the nonlinear interaction \(F\) are supposed to satisfy certain hypotheses.
scientific article; zbMATH DE number 7350277
scientific article